(Completing the Square) Quadratics Linear graphs y = mx+c Parallel Plotting graphs by substitution into equations Perpendicular Solving Quadratic & Linear Equations ax2+bx+c ax2+bx x2-y2 (x+#)(x+#) x(ax+b) (x+y)(x-y) Expanding Brackets Factorising Quadratic Graphs Expanding Double Brackets Solving Quadratics (Factorising) Solving Quadratic Equations Solving Linear Equations Solving Quadratics (Completing the Square) Solving Quadratics (Equation)

x x 2 x x2 2x 3x 6 x2 + 5x + 6 x a 6 a a2 6a 4 4a 24 a2 + 10a + 24 Return to main page x x 2 x x2 2x 3x 6 x2 + 5x + 6 x a 6 a a2 6a 4 4a 24 a2 + 10a + 24 x c 4 c c2 4c 3 3c 12 c2 + 7c + 12 x r 7 r r2 7r 2 2r 14 r2 + 9r + 14 x y 5 y y2 5y 3 3y 15 y2 + 8y + 15 x t 3 t t2 3t 8 8r 24 r2 + 11r + 24 x k 2 k k2 2k 6 6r 12 k2 + 8k + 12 x d 2 d d2 2d 8 8d 16 d2 + 10d + 16 x f -2 f f2 -2f 5 5f -10 f2 + 3f + -10 x g -4 g g2 -4g 2 2g -8 g2 + -2g + -8 x e -2 e e2 -2e 1 e -2 e2 + -e + -2 x b -1 b b2 -b 4 4b -4 b2 + 3b + -4

Expanding Double Brackets Return to main page 27d2 - 42d + 8 18q2 – 54q + 16 x2 - 4 t2 - 16 t2 + 4t + 4 9t2 + 30t + 25

Factorising Quadratics Return to main page Examples Factors of 10 Factors of -10 10 5 1 -10 -1 10 -5 -2 5 x 5 x -5 x 2 x2 x 2 x2 That add to 7 7x - 3x 10 - 10 (x +2)(x+5) (x +2)(x-5) That add to - 3 Factors of 15 15 -1 -15 3 5 - 3 -5 x -5 2x2 2x -3 - 10x - 13x - 3x 15 (2x -3)(x-5) Whereby add to - 13

Factorising Quadratics Return to main page (2x -3)(x+5) (5x +2)(x+1) (3x -2)(x+1) (5x -1)(x+7) (5x +1)(x+2) (5x -3)(x+2) (3x +8)(x-3) (2x +3)(3x-4) (x +1)(x+3) (x +1)(x+10) (x -2)(x-10) (x -1)(x+7) (x +3)(x-5) (x +2)(x+9) (x -2)(x-11) (x -4)(x-8) (x +5)(x-7) (x +8)(x-9) (x-11)(x-11) (x +4)(x-6) (x +3)(x-3)

Solving Linear Equations Return to main page Solving Linear Equations c= x= d= a= r= t= j= h=

Solving Quadratics (Factorising) Return to main page Solving Quadratics (Factorising) (x +5)(x+2) = 0 x +5 = 0 x +2 = 0 Making the equation x2 +7x + 10= 0 Means where does the graph of y = x2 +7x + 10 cross the line y = 0 Where does the graph of y = x2 +7x + 10 cross the line x = 0 / y axis

Solving Quadratics (Factorising) Return to main page Solving Quadratics (Factorising) x = 0 x = 4 s = 0 s = -2 g = 0 g = 2½ x = 0 x = -3⅓ h = 0 h = 4½ x = -3 x = -4 s = 3 s = 4 g = -4 g = 3 x = 4 x = -2 h = 2 h = 4 x = -1 x = -4 k = -2 k = -6 v = 2 v = 4 f = 3 f = 12 h = 2 h = -6 1) x = 0 x = 5 5) k = 0 k = 8 9) v = 0 v = 7 13) t = 0 t = 1½ 17) h = 0 h = 2⅓ 2) a = 0 a = -6 6) x = 0 x = -12 10) q = 0 q = 5 14) k = 0 k = - 2½ 18) c = 0 c = 4½ m = 7 m= - 8 a = - 3 a = - 4 x = -3 x = - 6 q = 3 q = 5 k = 4 k = 12 c= 4 c = -6

Solving Quadratics (Completing the Square) Return to main page Principle (x +3)(x+3) = (x+3)2 x x 3 x x2 3x 3x 9 x2 + 6x + 9 This sum of x terms will always be double the number being squared 6x Going the other way – to complete the square x2 + bx + c (x+b)2 2 Solve: x2 +8x + 10= 0 (x+4)2 – 16 + 10 = 0 This is required because (x+4)2 = x2 +8x + 16 so we need to subtract the 16 to preserve the value of the equation (x+4)2 - 6 = 0 (x+4)2 = 6 x = -4 ⁺⁄₋√6 x+4 = √6 x+4 = ⁺⁄₋√6

Solving Quadratics (Completing the Square) Return to main page What does completing the square tell us about the graph of y = x2 +8x + 10 y = (x+4)2 - 6 Because (x+4)2 is squared it is always positive, the smallest value it can have is 0 Therefore the smallest possible value of y is -6 This happens when x is -4 The equation of the line of symmetry is: x = -4

Solving Quadratics (Completing the Square) Return to main page In summary y = x2 +8x + 10 ½ of 8 10 - 42 y = (x+4)2 - 6 The equation of the line of symmetry is: x = -4 The minimum y value -6 Coordinates of the minimum (-4 ,-6)

Solving Quadratics (Completing the Square) Return to main page For each equation identify the equation for the line of symmetry and the minimum

Solving Quadratics (Equation) Return to main page Solving Quadratics (Equation)

Plotting graphs by substitution into equations Return to main page

Plotting graphs by substitution into equations Return to main page 18 4 0 -2 4 10 18

Plotting graphs by substitution into equations Return to main page 6 0 -4 -6 -4 0 6 14

Plotting graphs by substitution into equations Return to main page 18 9 -3 -6 -6 -3 2 9 18

Sketching graphs by using all the information Return to main page Sketch the graph of the following equation: y = x2 + 5x + 4 Using the line of symmetry Crosses the y-axis at 4 Factorises to: (x+1)(x+4) Completing the square: (x+2½)2-2¼ = 0 Solutions for: (x+1)(x+4)=0 Are x = -1 x = -4 Line of symmetry x = -2½ minimum coordinates (-2½, 2¼) y = x2 - 10x + 25 y = x2 + 4x - 21